I'm mainly interested in soccer related statistics. There are quite a few different API's relating to soccer, but most of them are commercial and far, far out of my price range. I've looked at DBpedia, but a lot of their data is quite out of date.

I would recommend xmlsoccer.com (xmlsoccer.com) - it’s pretty cheap (€10 per month) so it's not completely free, but it includes livescores, fixtures, results, odds, statistics etc. It also enables you to code for FREE against it using the Scottish Premier League as “demo-league” so you have unlimited time to develop your project. Full disclosure: I created it as a result of sitting in the same situation as you, and was a bit annoyed of lacking a reasonably priced API that was also fast and reliable. Best regards Michael
– MichaelAug 5 '17 at 20:09

Check out the Equity in Athletics Disclosure website http://ope.ed.gov/athletics/ supported by the Office of Postsecondary Education (OPE), U.S. Department of Education (ED).

The downloadable data sets includes data such as: Participants by sport and gender, coaching staff and salaries, revenues and expenses by sport and "game day" and recruiting, and other supplemental information.

This is interesting, and worth sharing... but I couldn't find anything that describes MLB's license or copyright on the data. It sounds like they've tolerated access to it for a few years.
– Joe GermuskaOct 14 '13 at 14:54

I use football-data.co.uk for 22 leagues, where a lot is provided, but some stats only for top leagues. If you want power rankings for particular teams, then football picks is also the option, but no API at the moment available. Of course you can combine knowldedge from livescore sites and then you can for instance build dataset of your own. Or you can go for opta sports and pay few thousands euros a month and you are perfectly equipped with everything!
From what I tried, another APIs that should have been for free are either not working or not for free anymore.
I hope this helps.

YouFoot offers both huge amounts of data as well as API and apps (web and mobile).
Currently access to our APIs are on a request basis. We look to make them public soon.

YouFoot is not only about distributing data but also about empowering people to collect complex sets of football data easily using only web or mobile apps.
Our match commentary apps allow you to produce stats and live test commentary in over 15 languages simultaneously.

This is an API for the World Cup(2014,2018) and Women's World Cup(2015,2019) that scrapes current match results and outputs match data as JSON. No guarantees are made as to its accuracy, but we will do our best to keep it up to date. For example responses, including events such as goals, substitutions, and cards, see the GitHub page.

Thank you for your interest in this question.
Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).